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The concept of logarithmic representation of infinitesimal generators isintroduced, and it is applied to clarify the algebraic structure of bounded andunbounded infinitesimal generators. In particular, by means of the logarithmicrepresentation, the bounded components can be extracted fromgenerally-unbounded infinitesimal generators. In conclusion the concept ofmodule over a Banach algebra is proposed as the generalization of Banachalgebra. As an application to mathematical physics, the rigorous formulation ofrotation group, which consists of unbounded operators being written bydifferential operators, is provided using the module over a Banach algebra.

Theory of $B(X)$-module -Algebraic module structure of generally-unbounded infinitesimal generators-